Let $p$ and $q$ be any two propositions. Consider the following propositional statements.
$S_1: p \to q$ $S_2: \neg p \land q$ $S_3: \neg p \lor q$ $S_4 : \neg p \lor \neg q$
where $\land$ denotes conjunction (AND operation), $\lor$ denotes disjunction (OR operation), and $\neg$ denotes negation (NOT operation). Which one of the following options is correct?
(Note: $\equiv$ denotes logical equivalence)
Logical Equivalence of Propositional Statements
Understanding the Implication Operator
The implication statement $p \to q$ is true in all cases except when $p$ is true and $q$ is false.
This implication is logically equivalent to the disjunction statement $\neg p \lor q$. This is a fundamental rule in propositional logic.
Analyzing Statements $S_1$ and $S_3$
- Statement $S_1$ is given as: $p \to q$
- Statement $S_3$ is given as: $\neg p \lor q$
Establishing Equivalence
Based on the standard logical equivalence rule for implication:
$ p \to q \equiv \neg p \lor q $
Therefore, statement $S_1$ is logically equivalent to statement $S_3$.
$S_1 \equiv S_3$
Important Questions from Propositional Logic
- Assume that a Creative ($C$) person will Succeed ($S$) if the person is also Disciplined ($D$), but will not succeed otherwise. Now, consider the following statements:
(i) $C \land S \Leftrightarrow D$
(ii) $C \Rightarrow (S \Leftrightarrow D)$
(iii) $C \Leftrightarrow ((D \Rightarrow S) \lor \neg S)$
Which of the following options is correct? - Let $x$ and $y$ be two propositions. Which of the following statements is a tautology /are tautologies?
Which of the following logical implications are tautology?
A. $[\sim p \land (p \lor q)] \rightarrow q$
B. $\sim p \rightarrow (p \rightarrow q)$
C. $[p \land (p \rightarrow q)] \rightarrow q$
D. $\sim (p \rightarrow q) \rightarrow q$
Choose the correct answer from the options given below:
- Sentence $X$ is said to entail Sentence $Y$ if whenever $X$ is $TRUE$, $Y$ also must hold $TRUE$.
Which of the following statements is/are correct if $X$ entails $Y$? - Let P denote "She is intelligent" and let Q denote "She is happy."
Given below are the statements:
(a) If She is intelligent, then She is unhappy.
(b) She is neither intelligent nor happy.
(c) It is necessary to be not intelligent in order to be happy.
(d) To be not intelligent is to be unhappy.
Which of the following is correct Propositional expression for the above statements:
